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International Journal of Pharmaceutics 511 (2016) 488–504

Considerations on the quantitative analysis of apparent amorphicity ofmilled lactose by Raman spectroscopy

Samaneh Pazesha,*, Lucia Lazorovaa, Jonas Berggrenb, Göran Alderborna, Johan Gråsjöa

aDepartment of Pharmacy, Uppsala University, Uppsala, SwedenbRecipharm Pharmaceutical Development AB, Solna, Sweden

A R T I C L E I N F O

Article history:Received 17 May 2016Received in revised form 30 June 2016Accepted 1 July 2016Available online 7 July 2016

Keywords:Raman spectroscopyLactoseAmorphous contentSpectral data analysisPrincipal component analysis (PCA)Milling induced disorder

A B S T R A C T

The main purpose of the study was to evaluate various pre-processing and quantification approaches ofRaman spectrum to quantify low level of amorphous content in milled lactose powder. To improve thequantification analysis, several spectral pre-processing methods were used to adjust background effects.The effects of spectral noise on the variation of determined amorphous content were also investigatedtheoretically by propagation of error analysis and were compared to the experimentally obtained values.Additionally, the applicability of calibration method with crystalline or amorphous domains in theestimation of amorphous content in milled lactose powder was discussed.Two straight baseline pre-processing methods gave the best and almost equal performance. By the

succeeding quantification methods, PCA performed best, although the classical least square analysis(CLS) gave comparable results, while peak parameter analysis displayed to be inferior.The standard deviations of experimental determined percentage amorphous content were 0.94% and

0.25% for pure crystalline and pure amorphous samples respectively, which was very close to thestandard deviation values from propagated spectral noise.The reasonable conformity between the milled samples spectra and synthesized spectra indicated

representativeness of physical mixtures with crystalline or amorphous domains in the estimation ofapparent amorphous content in milled lactose.ã 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/).

Contents lists available at ScienceDirect

International Journal of Pharmaceutics

journa l home page : www.e l sev ier .com/ loca te / i jpharm

1. Introduction

The interest in the quantitative analysis of amorphous contentof pharmaceutical solids has increased considerably the last years.This stems from the facts that firstly, amorphous solids may beused to solve certain formulation problems, such as low solubilityof a drug, and, secondly, the processing of crystalline particles mayresult in, often undesired, formation of an amorphous phase(sometimes referred to as process induced disordering) that maystill have an impact on physical and chemical properties of thematerials and thus the final pharmaceutical product performance.Processing operations such as size reduction (Caron et al., 2011;Chamarthy and Pinal, 2008; Otte et al., 2012), compression(Kaneniwa et al., 1985) and dry mixing (Pazesh et al., 2013) hasbeen shown to induce the formation of thermodynamicallyunstable amorphous regions of predominately crystalline

* Corresponding author at: Department of Pharmacy, Uppsala University, UppsalaBiomedical Center, P.O. Box 580, SE-751 23 Uppsala, Sweden.

E-mail address: [email protected] (S. Pazesh).

http://dx.doi.org/10.1016/j.ijpharm.2016.07.0010378-5173/ã 2016 The Author(s). Published by Elsevier B.V. This is an open access arti

particles. Therefore, the quantification of low levels of amorphousmaterials in powders has become an important part of thedevelopment of pharmaceutical preparations and this paper hasbeen written in the context of this aspect.

Many analytical techniques may be used to quantify modest tohigh levels of amorphous content in a solid, including X-raypowder diffraction (XRPD), differential scanning calorimetry(DSC), solution calorimetry, isothermal microcalorimetry anddynamic vapour sorption. It is reported that the quantificationlimit of XRPD and DSC are greater than 5% (Shah et al., 2006),whereas solution calorimetry (Hogan and Buckton, 2000),isothermal calorimetry (Buckton et al., 1995) and dynamic vapoursorption (Mackin et al., 2002; Sheokand et al., 2014; Young et al.,2007) may enable the quantification of amorphous content of lessthan 1%.

Vibrational spectroscopy techniques, such as near- infraredspectroscopy (NIR) (Fix and Steffens, 2004; Hogan and Buckton,2001; Savolainen et al., 2007), mid-infrared spectroscopy (MIR)(Agatonovic-Kustrin et al., 2001; Bartolomei et al., 1997) andRaman spectroscopy (Niemelä et al., 2005; Susi and Ard, 1974;Taylor and Zografi, 1998), have gained increasing interest in past

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S. Pazesh et al. / International Journal of Pharmaceutics 511 (2016) 488–504 489

decades for the analysis of chemical and physical properties ofpharmaceutical solids. In a review, Strachan et al. (Strachan et al.,2007) give an overview of a number of such applications of theRaman technique, including the analysis of crystallinity of a solid.The limited water sensitivity of the Raman signal is a majoradvantage of this technique in the study of amorphous solids sincewater is frequently present as moisture in the amorphous phase.

A number of papers on the quantification of amorphous contentof pharmaceutical substances by Raman spectroscopy have beenreported in the literature. An early paper on the subject wasauthored by Taylor and Zografi (Taylor and Zografi, 1998) whichwas followed by other papers dealing with both quantification ofamorphous content of drugs (e.g. Heinz et al. (Heinz et al., 2009),Mah et al. (Mah et al., 2015)) and excipients (Savolainen et al.(Savolainen et al., 2007) and Fix et al. (Fix and Steffens, 2004)).

According to literature, there are two ways of tackling thequantitative analyses of a Raman spectrum. The first group ofmethods is based on peak analysis, using peak variables such asamplitudes and areas or combinations thereof. In the second groupof methods, the spectral values are used as they are, irrespectivelyof what spectral feature they may belong to, and subjected tomultivariate analyses, such as principal component analysis (PCA)and classical least square analysis (CLS). Before the quantitativeanalysis, a raw Raman spectrum is often pre-processed in order toremove unwanted baseline and intensity variations in the Ramanspectrum due to fluorescence and macroscopic variations insample structure. The choice of pre-processing strategy may becrucial in order to obtain robust and accurate quantitativeinformation from Raman spectra. Nevertheless, there are onlyfew papers discussing the quantification of amorphous content of apharmaceutical substance that also reports on the effect of the pre-processing procedure of raw Raman spectra for the out-come ofthe analysis. Hence, this is an aspect of the spectrum analysis thatdeserves more attention.

The standard procedure used in the literature to transform aRaman spectrum of a sample of milled particles into amountdisordered content, is to use a standard curve obtained fromspectra of reference samples. These reference samples are typicallyphysical mixtures of completely crystalline and completelyamorphous particles of known proportions, i.e. two-state systemswith domains that are either crystalline or amorphous. Therepresentativeness of such a two-state system for the physicalstructure of milled particles that are considered partiallyamorphous may however be questioned. Two conceptions areused in the literature (Chamarthy and Pinal, 2008; Luisi et al., 2012)to describe the physical nature of partially amorphous particles.Firstly, particles can be considered as a one-state system where thedegree of disorder depends on the concentration of defects or thesize and orientation of crystallites forming the particle. Secondly,particles can be considered as a two-state system where the degreeof disorder depends on the proportion of amorphous andcrystalline domains. The physical form of milled particles is thusan intricate issue as the exact physical nature of the disorder inmilled samples is not obvious. Thus, a careful interpretation of thedetermined percentage amorphous content must be seen as anapparent measure and will at least partly be a relative measure ofthe degree of disorder rather than an absolute measure ofamorphicity. The term apparent amorphicity or apparent amor-phous content is used in this paper to acknowledge the fact that thequantitative analysis of a disordered solid results in a singlepercentage value although the exact nature of the solid is unknownand possibly complex.

The overall aim of this study is to address the question of thepossibility to derive physically sound values of the apparentamorphous content for milled powders with a special reference tothe importance of the combination of pre-processing and

quantification methods for the determination of amorphouscontent in a powder. Lactose, one of the most commonpharmaceutical excipients, is used as model material in the study.The overall aim was sub-divided into three sub-objectives. The firstobjective was to evaluate the effect of a comprehensive series ofpre-processing methods combined with a series of quantificationmethods for the determination of fraction of apparent amorphoussolid in a powder. The quantification methods could be divided intotwo main approaches; either based on peak parameter analysis ormultivariate analysis (as illustrated in Figs. 1 and 2). In this part ofthe study, a series of physical mixtures of crystalline andamorphous lactose was used. The second objective of the studywas to evaluate how spectral noise influenced the variation indetermined fraction of amorphous content for the physicalmixtures under investigation. This can be done by simulations,but, if possible, a derivation of an analytical expression willfacilitate this evaluation. In this paper, an expression forpropagation of errors for the combination of linear backgroundand PCA analysis was derived. The third objective was to determineand compare the apparent amorphous content of milled powdersof lactose, using the same pre-processing and quantificationmethods as were used in the assessment of amorphous content(Figs. 1 and 2), and to evaluate the representativeness of thecalibration method in the estimation of apparent amorphouscontent.

2. Materials and methods

2.1. Materials and sample preparation

Crystalline a-lactose monohydrate Pharmatose1 200 M, (DFEPharma, the Netherlands) was used as received. Amorphouslactose was prepared from a 10% (w/w) aqueous solution ofa-lactose monohydrate, using a Büchi Mini Spray Dryer B-290Advance (Büchi Labortechnik AG, Flawil, Switzerland). Theaspirator rate was set to 100%, the spray gas flow to 40 L/minand the feed pump rate to 4.0 mL/min. The inlet and outlettemperatures were maintained at 155

�C and 98

�C respectively.

Ball milling of crystalline lactose was performed in a planetaryball mill (PM 100, Retsch, Germany) at 25

�C and 30 � 3% relative

humidity (RH) with rotation speed of 400 rpm. 1 g of lactose wasmilled in a stainless steel milling jar of a volume of 12 cm3 with 50stainless steel balls with a diameter of 5 mm corresponding to aball to powder mass ratio of 25:1. In order to prepare lactosesamples that could represent low, intermediate and high degree ofapparent amorphous content of lactose by the ball mill used in thisstudy, a series of pre-trials was conducted. Based on these pre-trials, milling times of 10, 300 and 1200 min were used for samplepreparation. For the two longer milling times, a combination ofmilling periods and pause periods was applied to allow cooling ofthe jar and thus minimize heating of the sample, i.e. after eachmilling period of 20 min a pause period of 5 min was used.

For standard curves, physical mixtures with 0%, 5%, 10%, 15%,30%, 50%, 70%, 85%, 90%, 95% and 100% (w/w) amorphous contentwith total weight of 2 g were prepared by combining spray driedlactose powder, representing 100% amorphous lactose, andPharmatose powder, representing 100% crystalline lactose. Con-stituents of each mixture were added by geometrical dilution andmixed in a mortar with a pestle at 25 �C and 30 � 3% RH.

2.2. Experimental methods

2.2.1. X-ray powder diffractionPowder X-ray analysis were performed using a Bruker D8

Advance diffractometer equipped with a position sensitivedetector (PSD), LynxEye (Bruker AXS, Inc., Madison, WI, USA)

Fig. 1. Flow chart of different pre-processing and quantification approaches using multivariate analysis of Raman spectra.

490 S. Pazesh et al. / International Journal of Pharmaceutics 511 (2016) 488–504

and CuKa anode; l = 1.5406 Å. The sample was placed in a sampleholder and scanned at 40 kV, 40 mA from 10� to 60� 2u using ascanning speed of 0.2 s and a step size of 0.016�. A motorizedprimary and secondary divergence slit with 0.4� and 2.48� wereused. The diffractogram were collected using DIFRACT.SUITsoftware. The experimental diffraction patterns were compared

Fig. 2. Flow chart of different pre-processing and quantification a

with theoretical patterns which were based on the data from theCambridge Crystallographic Data Centre (CCDC).

2.2.2. Raman spectroscopyRaman spectra was collected using Raman SLSR-ProTT analyser

(Enwave Optronics., Irvine, CA, USA), equipped with TE cooled CCD

pproaches using peak parameter analysis of Raman spectra.


detector and a laser source with an excitation wavelength of785 nm. A reference standard (ECS-I, provided by the manufacturertogether with the Raman equipment) was used to calibrate thespectrometer each time the instrument was switched on. Themeasurements were performed at room temperature in a darkroom to minimize the influence of external light. In order toidentify potential procedure factors that could be of importance forquantification of amorphous fraction of lactose by Raman, a seriesof pre-trials was conducted. These included sample preparation(compact or powder), rotating or stationary sample holder, laserpower, measurement time and sample to laser probe distance.Based on the pre-trial outcome, the following procedures weredecided to be used: Compacts containing 400 � 2 mg of lactosesamples were prepared by powder compression in a single punchpress (Korsch EK0, Korsch, Germany) at an applied pressure of100 MPa, using circular flat-faced punches with a diameter of11.3 mm. The compressed sample (3 � 0.5 mm thick) was thenplaced in a rotating sample holder which was positioned under thelaser probe with a fixed distance of 7 mm. The rotation speed was�30 rpm and the radius of the circle swept by the laser spot wasaround 3 mm. Laser power and integration time was tuned in orderto avoid excessive heating of the sample. The laser output powerwas set to 30 mW and the integration time was set to 3 s over theRaman shift range of 100 cm�1 to 2200 cm�1. Each spectrumconsisted of 100 scans, resulting in average measurement time of5 min per measurement. The diameter of the laser beam at focus(the illuminated spot) was approximately 0.3 mm.

2.3. Pre-processing methods

2.3.1. Baseline correction and normalization methodsThe measured spectra may be regarded as consisting of the

Raman shifted signal superimposed on a background offset, mainlyoriginating from fluorescence. Further slight signal intensityvariations are noted due to variations in the preparation of thesamples. Thus to make the different Raman spectra comparable,the spectra have firstly, to be corrected by subtracting a baselinethat ideally should be equal to the fluorescence signal profile oranother offset reflecting the baseline, and secondly, to benormalized by the intensity. Basically, normalization is performedby dividing the intensity at each wavenumber of a spectrum with aconstant, specific for the spectrum and normalization method.

Regarding the background, one is, in practice, more or lessreferred to qualified guesses to at least come close to the exactshape of the real fluorescence profile. The problem is well known inmost types of spectroscopy, e.g. NMR spectroscopy, massspectroscopy, UV spectroscopy and different types of X-ray basedspectroscopies. On the other hand, if the systematic error to a largedegree is the same in all different sample spectra that are to becompared, the errors will in some degree cancel out each other ifspectra are used for quantification.

Regarding the normalization, typical strategies has been tonormalize to a certain peak amplitude or the area covered by thesignal in a certain wavenumber section. As shown in thesucceeding sections, also other strategies to handle the back-ground and normalization problem have been used.

In this investigation, low wavenumber region of spectrum (319cm�1 to 491 cm�1) was chosen for quantification analysis. Thewavenumber region used in Raman spectrum analysis may affectthe sensitivity of measurement dependent on the type ofquantitative analysis to be performed e.g. (Mah et al., 2015).Within the scope of the presented study, the exact choice ofwavenumber region is less important and a region that has earlierbeen used in Raman spectrum analysis of lactose sample wasarbitrarily chosen (Niemelä et al., 2005).

2.3.2. Straight baseline methods and higher polynomial baselinemethod approaches

In the straight baseline method, hereafter denoted method A,the simplest baseline approach was employed, where a straightline between the spectrum intensity values at 319 cm�1 and491 cm�1 was used as baseline. The corrected spectrum wascalculated by subtracting the spectra intensity values by thestraight line values at the respective wavenumber. The baselineapproach using a straight line to correct the spectrum was alsoapplied in straight baseline with Savitsky-Golay filtered endpoints,hereafter donated method B. Instead of using the recordedintensities at 319 cm�1 and 491 cm�1 when calculating the linearbaseline, the corresponding Savitsky-Golay filtered (3rd degree,n = 9) intensity values at 319 cm�1 and 491 cm�1 was used. Apossible advantage with the method B could be the diminishing ofnoise effects on the baseline that subsequently could propagateinto the determined amorphous fraction.

In polynomial method, here denoted method C, a modificationof the Lieber et al. (Lieber and Mahadevan-Jansen, 2003) methodwas done. Lieber proposes an iterative algorithm (Polyfit) thatsuggests a separation of regions of the spectrum that could beregarded as pure background (due to fluorescence) from regionsthat is due to vibrational effects. A polynomial is fitted to themeasured intensity in the background sections of the spectrum.The polynomial (in this work a 3rd degree polynomial) was fittedto the wavenumber region of the spectra that was to beinvestigated (in this work from 250 cm�1 to 540 cm�1). Portionsof the spectra that are positive relative the fitted line and above thenoise level are interim regarded as vibrational peaks and removedin a new polynomial fit. This procedure is repeated untilconvergence is reached and the then obtained polynomial is usedas the baseline. The calculations were carried out on an EXCELspread sheet linked to an in house developed visual basic macroprogram. The original method suggested by Lieber et al. does notconsider the noise level when determine vibrational regions of thespectra.

In methods A, B and C, the spectra, after baseline subtraction,were normalized by dividing the spectrum with the area under thecurve, numerically determined by the trapezoidal method.

2.3.3. Standard normal variate (SNV)In the standard normal variate method (SNV), here denoted

method D, the spectrum is transformed according to Eq. (1):(Barnes et al., 1989)

ySNV;i ¼zi� < z >

szð1Þ

where zi is the measured intensity at wavenumber i, < z >¼ 1n

Pzi,

i.e. the mean in the interval, and sz ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

zi� < z >ð Þ2=ðn � 1Þq

, i.e.the standard deviation in the interval. The summations are carriedout over all wavenumber points in the interval and n is number ofwavenumber points.

By the subtraction of the mean in the numerator, considerationfor the various backgrounds of the spectra of the samples wascarried out, however not necessarily resulting in a properbackground removal as in the methods described above. Therefore,in this way pre-processed spectrum is not usable for Peakparameter based quantification methods (see below). Further,the normalization is carried out by the division by the standarddeviation. In this work the method was applied to the wavenumberregion from 319 cm�1 to 491 cm�1.

2.3.4. Extended multiple scattering correction (EMSC) methodsIn extended multiple scattering correction (EMSC), hereafter

denoted method E, the spectrum is regarded being described


according to the following model (Eq. (2)): (Martens et al., 2003)

zj ¼ aj þ bj �zam þ zcryst

2

� �þ hj � zam � zcryst

� �þ dj � x þ ej � x2 ð2Þ

where zj is the vector representation of the spectrum for sample j,zcryst and zam are the vector representation of the average spectrumfor the pure crystalline and amorphous samples respectively and x

and x2 are the vectors of wavenumbers and squared wavenumbersrespectively. Further aj, dj and ej is the coefficients of thewavenumber dependent offset polynomial of degree 2, bj is thescaling (for normalization) and hj is linked to the amorphicity. Thecoefficients aj, bj, dj and ej was determined by multiple linearregressions and used to determine an offset and normalizedcorrected spectrum, yEMSC;j according to Eq. (3):

yEMSC;j ¼zj � ðaj þ dj � x þ ej � x2 Þ

bjð3Þ

In this work, the method was applied to the wavenumber regionfrom 319 cm�1 to 491 cm�1. By the bj coefficient, normalization isperformed and by the aj, dj and ej coefficients, this methodcompensates for offset effects such as fluorescent effects, but, insimilarity with the SNV method, this is not necessarily resulting ina proper baseline removal. Therefore, in this way pre-processedspectra is not either usable for peak parameter based quantifica-tion methods (see below). To obtain a proper baseline removal, andwith that pre-processed spectra appropriate for peak parameterbased quantification methods, a second approach of EMSC wasapplied, here denoted method F. In Eq. (2) the spectra vectors zcrystand zam, were replaced by corresponding vectors of backgroundcorrected and normalized spectra (according to the method A).Thereafter a normalized corrected spectrum yEMSC;j was obtainedaccording to Eq. (3).

2.3.5. Pre-processing method provided by PeakFit softwareA baseline algorithm offered within the PeakFit software was

also used, here denoted method G. Sections of the spectrum in therange 315–496 cm�1 were detected, where the second derivative ofthe data is both constant and zero, which were regarded to besections where the baseline is not superposed by spectral features.A linear baseline was fit to these sections and subtracted from thespectrum, where after the data were normalized to unit area.

2.4. Quantification methods

2.4.1. Principal component analysis (PCA)Principal component analysis (Simca software, Umetrics,

Sweden) was conducted on data pre-processed in the differentways described above (method A-F). The wavenumbers weretreated as variables and the different physical mixtures of lactoseas observations. Before the extraction of principal component, thedata were centered scaled (Ctr). The other standard scalingmethods such as unit variance (UV) and non-scaling were brieflyinvestigated, but showed a slightly worse performance regardingquantification and were therefore suspended in advantage for thecentered scaling. Regarding the observations included in the PCAtwo main approaches were tested, where either the PCA wasapplied on the spectra of all physical mixtures or applied just onthe spectra of pure crystalline and pure amorphous samples. Theresults of the analyses did show only minute differences in theloading-values and therefore the latter, simpler, approach waschosen.

Using the PCA loadings, the score (t) for the 1 st principalcomponent (PC1) of each physical mixture and milled samplespectra (after centering) was calculated. The fraction of amorphous

content (fam) could then be obtained from the score according toEq. (4):

f am ¼ tc � tðC � 1Þt � ðC � ta � tcÞ ð4Þ

Where tc and ta is the averages of the score for PC1 of purecrystalline sample, and pure amorphous sample respectively and Cis the estimated integrated intensity ratio between pure amor-phous and pure crystalline spectra. The derivation of Eq. (4) andthe procedure to determine C is given in Appendix A. The scoresand quantification calculations were carried out on an EXCELspread sheet where in the determination of C, the with thesoftware provided solver function was used.

2.4.2. Analysis by classical least squares (CLS)The measured spectra (in the selected wavenumber region of

Raman spectra pre-processed as in A-F) were expected to be thesum of amorphous and crystalline spectra weighted by theproportions, which gave the following relation (Eq. (5)):

ysynth ¼ ð1 � f amÞ � yc þ f am � C � yað1 � f amÞ þ f am � C

ð5Þ

where fam is the proportion of amorphous content of the measuredsample, yc and ya are the vector representations of the normalizedcrystalline spectrum and the normalized amorphous spectrumrespectively and ysynthis the vector representation of the resultingsynthesized spectrum, mimicking the normalized measuredspectrum. The derivation of Eq. (5) and the procedure to determineC is given in Appendix A. The fraction of amorphous content (fam)for a sample of unknown composition could be obtained by anordinary curve-fitting procedure, i.e. by minimizing the sum ofsquares (Eq. (6)):

Xni¼1

yi � ysynth;i� �2

ð6Þ

where yi and ysynth;i are the i-th components of the samplespectrum and ysynth respectively, according to the equation above(Eq. (5)). The calculations and curve fitting (by the Levenberg-Marquardt method) were carried out on am EXCEL spreadsheetlinked to a visual basic macro.

2.4.3. Peak parameter analysis by PeakFitThe software PeakFit (version 4.12, Systat Software, Inc., San

Jose, USA) was used to extract peak parameters from the rawRaman spectra (Fig. 3). Prior to the peak fitting, several pre-processing methods were applied (A, B, C, F, G described in the pre-processing section), where after the pre-processed data weresmoothed by Savitzky-Golay algorithm for noise reduction.

The number of peaks to be examined in selected wavenumberregion (315–496 cm�1) was set to seven (Fig. 4), based on theoutlook of the spectra in the region. An initial manual fit of theseven peaks (assumed having Voigt profiles) was done, followed byan automated fit using the Levenberg-Marquardt algorithm, wherethe peak amplitudes, peak centers, full peak width at half-maxima(FWHM) and full base width for each peak were obtained. Thefactors used as indicators for the goodness of the fit werecoefficient of the determination and the F-statistic for the fit.

In order to sift out the most potent parameters for quantifica-tion, a pre-investigation of the correlation or anti-correlationbetween the obtained peak parameters and fraction amorphouslactose was performed. Based on calculated correlation coefficientsbetween fraction amorphous lactose and the different peakparameters and by visual inspection of the corresponding graphs,it was concluded that peak areas and peak amplitudes showed thebest correlation or anti-correlation to fraction amorphous lactose.

Fig. 3. Raw Raman spectra of pure crystalline a-lactose monohydrate (the black line) and pure amorphous lactose (red dashed line) before baseline correction.


Different ways using the peak data to obtain measures forquantification were further investigated by single peak amplitudes(I), summed peak areas (AS), the ratios of peak areas (AR) and ratiosof peak amplitudes (IR).

From such measures, the fraction of amorphous content (fam)could then be obtained according to Eq. (7):

f am ¼ rc � rðC � 1Þr � ðC � ra � rcÞ ð7Þ

where rc is the average of the I, AS, AR or IR measure of purecrystalline sample, ra is the average of the I, AS, AR or IR measure ofpure amorphous sample, r is the I, AS, ARor IRmeasure of the sampleunder investigation and C is the estimated integrated intensityratio between pure amorphous and pure crystalline spectra. Thederivation of Eq. (7) and the procedure to determine C is given inAppendix A. It should here be mentioned that applying Eq. (7) onpeak area ratios and peak intensity ratios, the obtained quantifi-cation methods become almost identical to the quantificationmethod suggested by Taylor and Zografi (Taylor and Zografi, 1998)

Fig. 4. Spectra of crystalline lactose (A) and 100 w/w% amorphous lactose (B) in the selectSavitsky-Golay filtered (quadratic, window size = 9). The single peaks (coloured lines)

amount of amorphous lactose are labelled with red numbers and peaks co-variating w

(Kontoyannis et al. (Kontoyannis et al., 1997), and Roberts et al.(Campbell Roberts et al., 2002)).

2.5. Quantification of milled samples

Milled lactose samples were analyzed by all above describedquantification methods and the amount of apparent amorphousfractions (w/w%) after 10 min, 300 min and 1200 min milling weredetermined.

2.6. Calculation of the percentage error due to propagated spectralnoise

To estimate the spectral noise level by the variance, tenrepeated measurements was performed on the same rotatingsample, with duration of 30 s of, i.e. 1/10 of the time of the standardmeasure time in this work. The variance was then calculated ateach wavenumber and pooled, where after the variance of a 300 smeasurement was estimated by dividing the pooled variance with

ed wavenumber region (black lines). The spectra are pre-processed by method A andare extracted with aid of the PeakFit software. Peaks co-variating with increasingith increasing amount of crystalline lactose are labelled with black numbers.


the factor 10. To simplify the succeeding analysis, the variance ofthe noise was assumed being constant ðs2

z Þ within the wave-number region used in the analysis. The effect of the spectral noisepropagated into the standard deviation ðsf Þ, of determinedamorphous content was then calculated according to Eq. (8)(see Appendix B for derivation):

sf ¼C � ta � tcð Þ

C t � tað Þ � ðt � tcÞ½ �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2num

A2 þ N2s2area

A4

sð8Þ

where

s2num ¼ s2

z �Xn�1

i¼1

p2i þ2s2

z

Dx2range

Xn�1

i¼1

pi � xi

!2

ð9Þ

s2area ¼ n � 1ð Þ � s2

z � Dxð Þ2 þ s2z

2� ðn � 1Þ2 � Dxð Þ2 ð10Þ

N ¼Xn�1

i¼1

pi � zi � k �Xn�1

i¼1

pi � xi ð11Þ

and were xi is the wavenumber difference to the center-wave-number of the spectral portion under analysis, Dx = xi� xi-1, A is thenorming area under the baseline corrected spectrum and pi is theloading for PC1 for wavenumber i.

Fig. 5. X-ray powder patterns of crystalline a-lactose monohydrate, amorphous lactoseintensity diffractograms of 1200 min milled lactose (blue) and spray dried lactose (pur

3. Results

3.1. Characterization by XRPD

The X-ray diffraction pattern of a-lactose monohydrate asreceived corresponded to its theoretical diffraction pattern andconfirmed its crystalline nature. Spray dried lactose wascompletely amorphous, indicated by a “halo” pattern and theabsence of distinct diffraction reflections. Fig. 5 shows theexperimental X-ray diffraction pattern of the crystalline a-lactosemonohydrate, amorphous lactose and milled lactoses of millingtimes of 10 min, 300 min and 1200 min. For the milled samples, theintensity of peaks became lower and the peaks became slightlybroader and merged with increasing milling time.

3.2. Qualitative observations of relations between Raman peaks andamorphous content

The selected region of spectrum appeared to significantlydifferentiate between mixtures of various composition of crystal-line and amorphous lactose (Fig. 6). By the seven peaks found inthe selected region, the amplitudes and areas of peaks 5 and 6 wereco-variating with increasing amount of amorphous lactose inphysical mixtures and thus were referred to as “amorphous peaks”and on the contrary, amplitudes and areas of peaks 1, 2, 3, 4, and 7were co-variating with increasing amount of crystalline lactoseand thus were referred to as “crystalline peaks” (Fig. 4).

(spray dried) and milled lactose. The insert shows an expanded view of the weakerple).

Fig. 6. Raman spectra of physical mixtures containing merely 0, 10, 30, 50, 70, 90, and 100% amorphous lactose (w/w%). The spectra are pre-processed by method A andSavitsky-Golay filtered (quadratic, window size = 9). Black line indicates pure crystalline samples; red line indicates pure amorphous samples.


With increasing content of amorphous lactose, FWHM wasbroadening in peaks with amorphous character (e.g. peak 6), butwas generally quite stable in crystalline peaks (e.g. peak 7). Inaddition to peak broadening, peak overlapping and merging wasobserved with increasing content of amorphous lactose (Figs. 4 and6). Peak centra were also shifting to some extent, mostly in peak 3(from 375 to 383 cm�1), peak 6 (from 439 to 444 cm�1) and peak 4(from 396 to 402 cm�1) and to least extent in peak 1 (from 339 to

Table 1Summary of quantification analysis of amorphous fraction by different pre-processing

Pre-processing method A B

Nominal amorphous content (w/w%)

0 0.00 � 0.94 0.00 � 0.97

5 4.40 � 1.43 4.31 � 1.44

10 9.58 � 1.06 9.56 � 1.15

15 12.93 � 1.50 12.86 � 1.57

30 30.42 � 0.43 30.33 � 0.36

50 49.84 � 2.05 50.19 � 1.97

70 70.22 � 1.95 70.21 � 1.99

85 85.85 � 1.37 85.57 � 1.37

90 91.64 � 0.75 91.35 � 0.72

95 95.95 � 0.57 95.77 � 0.51

100 100.00 � 0.25 100.00 � 0.25

Average SD 1.12 1.12

Max SD 2.05 1.99

RMSD 1.02 0.95

MD 2.07 2.14

Raw spectra were pre-processed using methods A-F as described in method section. The pmax standard deviation (max SD), root mean square deviation (RMSD) and max deviat

336 cm�1), peak 2 (from 354 to 351 cm�1), peak 5 (from 419 to420 cm�1) and peak 7 (from 473 to 474 cm�1)

To be noted in the crystalline spectrum (Fig. 6), there are smallhumps on the high wavenumber flank of peak 6 and the lowwavenumber flank of peak 1, which probably are indications ofsmaller peaks merged with a stronger neighboring peak. Theseprobable peaks have not been considered in the peak fittingprocedure by the limiting of the number of peaks in order to reduce

methods of physical mixtures using principal component analysis (PCA).

C D E F

Calculated amorphous content (w/w%)

�0.01 � 1.57 �0.09 � 4.20 0.00 � 1.91 �0.01 � 1.912.33 � 2.00 7.30 � 1.67 5.02 � 1.94 5.02 � 1.947.27 � 1.10 9.10 � 1.65 10.50 � 1.22 10.50 � 1.2211.85 � 1.56 14.18 � 2.70 14.25 � 1.77 14.25 � 1.7728.16 � 0.80 30.37 � 0.13 30.88 � 0.33 30.88 � 0.3349.54 � 1.98 47.86 � 2.28 49.91 � 2.05 49.91 � 2.0571.99 � 2.23 67.54 � 2.85 69.10 � 2.11 69.10 � 2.1188.87 � 0.92 84.67 � 1.38 84.40 � 1.37 84.40 � 1.3794.02 � 0.58 90.77 � 0.55 90.41 � 0.67 90.41 � 0.6797.24 � 0.26 95.52 � 0.51 95.18 � 0.47 95.18 � 0.47

100.00 � 0.21 100.00 � 0.39 100.00 � 0.28 100.00 � 0.28

1.20 1.66 1.28 1.282.23 4.20 2.11 2.11

2.75 1.43 0.57 0.574.02 2.46 0.90 0.90

erformance of results was summarized in average standard deviation (average SD),ion (MD).

Table 2Summary of quantification analysis of amorphous fraction by different pre-processing methods of physical mixtures using the classical least squares method (CLS).

Pre-processingmethod

A B C D E F

Nominal amorphous content (w/w%) Calculated amorphous content (w/w%)

0 �0.01 � 1.22 �0.01 � 1.25 �0.01 � 1.16 �0.08 � 3.91 0.00 � 1.92 �0.02 � 2.475 4.55 � 1.30 4.48 � 1.32 2.95 � 1.77 5.66 � 1.64 5.07 � 1.95 5.59 � 1.8710 9.73 � 1.40 9.75 � 1.51 7.52 � 1.16 9.59 � 2.18 10.58 � 1.23 11.13 � 1.5915 13.69 � 1.78 13.66 � 1.87 12.80 � 2.15 14.83 � 2.81 14.37 � 1.78 15.63 � 2.1730 30.13 � 0.50 30.26 � 0.37 27.06 � 0.84 29.95 � 0.14 31.06 � 0.33 31.85 � 0.2250 49.81 � 1.96 50.53 � 1.89 48.54 � 2.09 48.34 � 2.03 50.17 � 2.03 51.30 � 1.9870 69.72 � 1.89 69.84 � 1.93 70.20 � 2.35 67.81 � 2.25 69.30 � 2.10 69.56 � 2.0785 85.53 � 1.47 84.02 � 1.59 86.45 � 1.08 83.09 � 1.46 84.49 � 1.38 84.06 � 1.4690 91.40 � 0.81 90.04 � 0.74 92.35 � 0.79 89.36 � 0.58 90.45 � 0.66 90.03 � 0.6795 95.50 � 0.67 94.28 � 0.52 95.75 � 0.42 94.00 � 0.57 95.15 � 0.46 94.52 � 0.49100 100.00 � 0.25 100.00 � 0.24 100.00 � 0.32 100.00 � 0.50 100.00 � 0.27 100.00 � 0.30

Average SD 1.22 1.20 1.28 1.64 1.28 1.39Max SD 1.96 1.93 2.35 3.91 2.10 2.47

RMSD 0.72 0.67 1.95 1.21 0.57 0.97MD 1.40 1.34 2.94 2.19 1.06 1.85

Raw spectra were pre-processed using methods A-F as described in method section. The performance of results was summarized in average standard deviation (average SD),max standard deviation (max SD), root mean square deviation (RMSD) and max deviation (MD).


the risk of over-fitting and un-robustness of the peak parameterdetermination.

3.3. Comparison of performance between the different pre-processingand analysis methods

The results by the determined percentages of amorphouscontent of lactose are shown in the Tables 1–6, where each tablereports values from a specific quantification analysis. Thedetermined amorphous percentage values and their standarddeviations are given for all physical mixtures (row wise) and for thedifferent pre-processing methods applicable (column wise).Summarized values for each pre-processing method are also givenby average and maximum standard deviation of the determina-tions as well as root mean square deviation (RMSD) and maximumdeviation (MD) from nominal percentage of the physical mixtures.

For the peak parameter quantification analyses, in order to get abetter overview, the results for the four best performing peakparameter measures for quantification are shown: i) single peakamplitudes by peak 6, ii) summed peak areas of peaks 5 and 6, iii)the ratios obtained by dividing the area of peak 6 by the sum of the

Table 3Summary of quantification analysis of amorphous fraction by different pre-processing m

Pre-processing method A B

Nominal amorphous content (w/w%)

0 �0.03 � 2.42 �0.02 �

5 4.05 � 2.56 3.93 �

10 9.39 � 0.92 9.34 �

15 11.80 � 1.13 11.66 �

30 29.79 � 0.60 29.87 �

50 48.88 � 2.11 48.68 �

70 71.22 � 2.76 71.94 �

85 87.51 � 1.46 88.26 �

90 92.85 � 0.97 93.46 �

95 96.73 � 0.45 97.50 �

100 99.94 � 0.79 99.94 �

Average SD 1.44 1.35Max SD 2.76 2.56

RMSD 1.88 2.29MD 3.20 3.46

Raw spectra were pre-processed using methods A-F as described in method section. The

max standard deviation (max SD), root mean square deviation (RMSD) and max devia

areas of peak 6 and 7 and iv) the ratios obtained by dividing theamplitudes of peak 7 by the amplitudes of peak 6. In order todetermine the accuracy of the quantification methods, validationcurves were constructed by plotting the nominal vs. determinedamount of amorphous lactose (w/w%) in the physical mixtures forall acquired calibration curves. Fig. 7 illustrates validation curve forthe pre-processing and quantification method here graded beingthe best performing, i.e. PCA preceded by the method A as pre-processing method.

3.4. The results from milled samples

The determined amounts of apparent amorphicity of milledlactose powder are shown in Table 7. The mean of apparentamorphicity values and their standard deviation are given for thethree milling times (row wise) and for the different pre-processingmethods applicable (column wise). Summarizing values for eachpre-processing method are also given by average and maximumstandard deviation of the determinations.

The obtained degrees of disordered lactose were inserted intoEq. (5) synthesising corresponding theoretical spectrum of

ethods of physical mixtures using analysis based on single peak amplitude of peak 6.

C F G

Calculated amorphous content (w/w%)

2.40 �0.06 � 3.22 �0.04 � 3.17 �0.03 � 2.742.56 1.87 � 3.44 1.72 � 3.17 4.54 � 2.520.93 6.45 � 2.37 7.84 � 0.95 10.86 � 1.231.12 9.06 � 1.66 10.67 � 1.38 11.33 � 1.990.69 26.38 � 2.09 30.99 � 0.56 28.86 � 0.762.06 48.77 � 2.93 49.87 � 1.86 48.23 � 2.241.81 75.13 � 2.13 71.27 � 0.45 72.26 � 1.941.45 91.36 � 0.70 86.98 � 1.43 88.73 � 1.661.06 96.79 � 1.11 91.00 � 2.20 94.48 � 1.860.45 99.44 � 0.77 97.34 � 0.38 99.02 � 0.510.78 100.08 � 0.98 99.97 � 0.76 99.99 � 1.24

1.91 1.38 1.52 3.44 3.17 2.52

4.77 2.29 2.60 6.79 4.33 4.48

performance of results was summarized in average standard deviation (average SD),tion (MD).

Table 4Summary of quantification analysis of amorphous fraction by different pre-processing methods of physical mixtures using analysis based on summed integrated peak areas ofpeak 5 and 6.

Pre-processing method A B C F G


0 0.00 � 0.96 0.00 � 0.95 �0.05 � 2.65 �0.02 � 1.86 �0.01 � 1.575 3.73 � 1.75 3.60 � 1.71 0.37 � 2.92 1.29 � 2.09 4.83 � 1.6510 9.64 � 0.92 9.59 � 0.97 6.18 � 2.94 7.81 � 1.44 12.15 � 1.2015 13.79 � 1.54 13.63 � 1.59 11.57 � 2.28 12.25 � 2.61 14.19 � 2.1630 31.12 � 1.23 31.96 � 0.85 27.68 � 1.81 33.32 � 0.33 27.97 � 1.1150 49.69 � 4.11 49.50 � 4.09 50.13 � 4.41 50.80 � 4.05 48.18 � 4.0170 69.51 � 2.43 70.03 � 1.64 71.50 � 1.69 69.38 � 0.26 71.55 � 1.4685 85.92 � 1.81 85.59 � 1.77 89.36 � 0.97 84.97 � 1.97 87.10 � 1.8990 91.69 � 0.94 91.43 � 0.80 95.01 � 0.94 89.43 � 2.66 93.32 � 1.0095 95.88 � 0.68 95.76 � 0.83 97.49 � 0.75 96.02 � 0.92 97.82 � 0.81100 99.98 � 1.38 99.98 � 1.36 99.99 � 0.65 99.98 � 1.53 99.98 � 1.43

Average SD 1.71 1.58 2.08 1.82 1.72Max SD 4.11 4.09 4.41 4.05 4.01

RMSD 1.02 1.11 3.43 2.10 2.00MD 1.69 1.96 5.01 3.71 3.32


Table 5Summary of quantification analysis of amorphous fraction by different pre-processing methods of physical mixtures using analysis based on ratios of integrated peak areas ofpeak 6 and summed areas of peaks 6 and 7.



0 0.00 � 0.73 0.00 � 0.71 �0.04 � 2.64 0.00 � 1.16 0.00 � 1.435 2.90 � 1.62 2.77 � 1.56 �0.65 � 2.44 0.95 � 1.72 4.45 � 1.6110 9.05 � 0.91 8.99 � 0.94 5.54 � 3.22 7.50 � 1.34 11.74 � 1.1715 12.46 � 1.27 12.32 � 1.32 8.78 � 1.53 11.31 � 2.07 13.52 � 2.2330 30.15 � 1.78 31.17 � 0.60 27.36 � 1.16 32.24 � 0.34 27.00 � 1.1350 49.59 � 4.63 49.31 � 4.59 49.94 � 4.38 50.39 � 4.97 49.74 � 4.7170 70.50 � 2.64 71.04 � 1.79 72.71 � 1.81 70.26 � 0.03 71.04 � 2.2185 87.29 � 1.60 86.95 � 1.51 91.22 � 1.40 86.10 � 1.93 87.53 � 1.6690 92.55 � 1.22 92.13 � 1.14 95.78 � 0.88 90.64 � 2.77 92.10 � 1.3995 97.17 � 0.82 96.69 � 0.81 98.47 � 2.02 97.52 � 0.93 97.35 � 0.69100 99.99 � 2.48 99.99 � 2.40 99.99 � 0.93 99.99 � 2.51 99.99 � 2.32

Average SD 1.83 1.58 2.09 1.79 1.87Max SD 4.63 4.59 4.38 4.97 4.71

RMSD 1.79 1.74 4.58 2.34 1.89MD 2.55 2.68 6.22 4.05 3.00


Table 6Summary of quantification analysis of amorphous fraction by different pre-processing methods of physical mixtures using analysis based on ratio of peak amplitudes of peak7 and peak 6.



0 0.08 � 2.05 0.09 � 2.07 0.07 � 3.25 0.17 � 2.88 0.08 � 2.405 4.32 � 2.37 4.24 � 2.40 1.93 � 3.51 2.15 � 3.08 4.30 � 2.4310 9.60 � 1.04 9.63 � 1.06 7.03 � 2.77 8.39 � 1.09 10.86 � 1.3815 11.97 � 1.06 11.95 � 1.09 9.56 � 1.32 11.23 � 1.56 11.46 � 1.7730 30.39 � 0.84 30.76 � 0.41 28.88 � 1.25 31.64 � 0.49 29.31 � 0.8050 49.05 � 2.33 48.95 � 2.30 49.19 � 1.97 50.08 � 2.41 48.94 � 2.2770 70.72 � 2.56 71.38 � 1.75 72.61 � 1.93 70.60 � 0.21 71.80 � 1.8685 86.79 � 1.24 86.66 � 1.21 89.12 � 1.09 86.33 � 1.33 87.07 � 1.2790 92.36 � 0.74 92.16 � 0.71 94.25 � 0.45 90.50 � 2.51 92.34 � 0.9995 97.14 � 0.74 96.88 � 0.73 97.78 � 1.09 97.40 � 0.82 97.26 � 0.64100 100.01 � 1.22 100.01 � 1.18 100.00 � 0.34 100.01 � 1.23 100.01 � 1.13

Average SD 1.43 1.29 1.71 1.50 1.49Max SD 2.56 2.40 3.51 3.08 2.43

RMSD 1.66 1.65 3.32 1.99 1.93MD 3.03 3.05 5.44 3.77 3.54

Raw spectra were pre-processed using methods A-F as described in method section. The performance of results was summarized in average standard deviation (average SD),max standard deviation (max SD), root mean square deviation (RMSD and max deviation (MD).


Fig. 7. Validation plot of the best performing quantification analysis, i.e. using pre-processing method A combined with PCA. A linear regression with an R2

value > 0.999 was obtained.


amorphous crystalline mixture. The spectra of milled lactose weresimilar to synthesized spectra with the corresponding amorphouscontent (Fig. 8) although some deviations however were obtainedfor 300 min and 1200 min milled samples relative the correspond-ing synthesized spectra.

3.5. Theoretical propagation of spectral noise analysis

For the crystalline sample spectrum, the estimated variancewas s2

z = 0.20 (sz = 0.45) in the 320 cm�1 to 490 cm�1 range. Thevariances did not show any significant correlation to eitherwavenumber or intensity spectral intensity and could thereforebe regarded as constant in the interval and the approximation of avariance being constant over all wavenumbers in the wavenumber

Table 7Determined apparent amorphous fraction in milled lactose samples using different pre

Pre-processing method A B C

Milling time (min) Calculated

PCA10 0.84 � 0.43 0.86 � 0.51 3.57 � 0.71

300 49.20 � 1.86 48.96 � 1.83 51.64 � 0.67

1200 82.36 � 0.81 82.22 � 0.82 83.79 � 1.98

CLS10 1.21 � 0.63 1.24 � 0.71 4.12 � 0.78

300 51.90 � 2.35 51.77 � 2.31 55.78 � 2.00

1200 83.88 � 0.97 83.81 � 0.97 85.33 � 2.57

Single peak amplitude

10 �0.04 � 0.84 �0.04 � 0.89 2.26 � 1.00

300 56.18 � 1.57 55.63 � 1.41 56.66 � 0.49

1200 92.38 � 1.04 92.40 � 1.11 85.54 � 2.77

Integrated peak areas anal10 0.14 � 0.42 0.16 � 0.51 2.68 � 0.78

300 47.19 � 1.78 47.01 � 1.63 42.68 � 0.67

1200 76.07 � 1.12 75.95 � 1.14 72.53 � 4.73

Ratios of integrated peak areas analyses of pe10 0.01 � 0.23 0.04 � 0.29 1.41 � 0.61

300 46.42 � 1.52 45.98 � 1.49 36.51 � 0.71

1200 75.40 � 0.73 75.36 � 0.76 76.67 � 5.01

Ratio of peak amplitudes analy10 0.28 � 0.72 0.29 � 0.78 2.71 � 0.85

300 52.81 � 1.70 52.63 � 1.57 53.29 � 0.63

1200 81.15 � 0.64 81.16 � 0.69 84.48 � 1.31

region was regarded appropriate. Putting the above determineds2z -value into Eqs. (8)–(11) gave a sf = 0.74% at 0% amorphous

content and a sf = 0.21% at 100% amorphous content. The standard

deviations of amorphous content in the amorphous and crystallineand amorphous samples was experimentally determined tosf = 0.94% and sf = 0.25% respectively (ncryst = namorph = 20). It could

thus be concluded that the theoretically determined standarddeviations values are in the same magnitude as, however slightlysmaller than, the corresponding experimentally determinedvalues.

4. Discussion

4.1. Physical mixtures

Amongst the four wavenumber regions identified showingdifferences between amorphous and crystalline lactose, the lowestwavenumber regions (319 cm�1–491 cm�1) were chosen. Thisregion might contain features emanating from lattice vibration(Carteret et al., 2009; Roy et al., 2013), which are expected todisappear in the amorphous state and therefore being a gooddiscriminator between crystalline and amorphous lactose.

The general broadening of peaks, especially peaks 5 and 6,observed in the amorphous lactose spectrum, could be attributedto an ensemble of slightly inequivalent sites. In crystalline lactose,the lattice arrangement of molecules gives that for each vibrationmode, the frequency is expected to be the same in molecules inequivalent lattice points. In amorphous lactose, on the contrary,the molecules are not regularly arranged, i.e. the surrounding ofevery molecule differs and with that a variation in frequencies ofeach vibration mode are expected leading to a broadening of theRaman peaks.

-processing methods and different quantification approaches.

D E F G

apparent amorphous fraction (w/w%)

1.64 � 0.54 2.37 � 0.15 2.36 � 0.15 –

52.76 � 2.11 51.54 � 1.88 51.54 � 1.88 –

77.94 � 1.23 81.83 � 0.79 81.83 � 0.79 –

2.68 � 0.50 2.40 � 0.15 2.89 � 0.30 –

54.44 � 2.40 51.88 � 1.86 55.45 � 2.49 –

78.01 � 0.98 82.08 � 0.78 84.46 � 0.96 –

analyses of peak 6– – 1.81 � 0.65 0.17 � 1.14– – 59.70 � 1.60 56.56 � 1.04– – 90.95 � 0.63 93.06 � 1.01

yses of peaks 5 and 6– – 1.88 � 0.15 �0.11 � 1.05– – 55.14 � 1.71 48.52 � 0.98– – 77.33 � 1.04 77.25 � 1.36

ak 6 and summed areas of peaks 6 and 7– – 1.34 � 0.11 �0.29 � 0.82– – 52.68 � 1.45 47.71 � 1.21– – 74.85 � 0.93 75.19 � 0.66

ses of peak 7 and peak 6– – 2.15 � 0.56 0.43 � 1.10– – 56.46 � 1.62 53.34 � 1.53– – 81.47 � 0.65 81.46 � 0.58

Fig. 8. Pre-processed Raman spectra of milled lactose (red dashed line) and thesynthesized spectra (the black line) with the corresponding amorphous content. (A)indicates spectra of 10 min milled lactose; (B) indicates spectra of 300 min milledlactose and (C) indicates the spectra of 1200 min milled lactose. Raman spectra ofmilled sample are pre-processed using pre-processing method A and quantified byPCA. The synthesized spectra are derived from the obtained amorphous content oflactose using PCA.


In this work measure of maximum deviation (MD) and RMSDhave been used as measures of accuracy and maximum andaverage standard deviation as measure of precision for the overallperformance. As discussed below, the precision of the amorphicitydetermination in mixed samples may be unrepresentatively highdue to sample heterogeneity. Therefore the precision measures(standard deviation) for the pure crystalline (0% amorphous) andpure amorphous (100% amorphous) samples i.e. where thesamples necessarily must be totally homogeneous with regardto composition, is more representative.

Assessing on the basis of these measures, the straight baselineapproaches (Method A and B) combined with quantification bymultivariate analysis were graded to be the best performing and

most reliable combination of pre-processing and quantificationmethods. EMSC (Method E and method F) combined withquantification by PCA actually performs even better than A andB regarding RMSE and maximum deviation, however the standarddeviation in the pure crystalline sample is twice the standarddeviation when Method A and B have been applied. To be noted, thepre-processing methods A and B and E combined with the CLS(synthetization) performs almost as well as when combined withPCA.

The methods A and B differed only by the Savitsky-Golayfiltering of the baseline determining end points and it could beconcluded that the filtering had just a minor impact on the baselineand on the quantification. The worse performance using the pre-processing method C (polyfit) may be explained by as well toomuch flexibility of the curvature of the baseline polynomial, as toomuch flexibility in the choice of data points used for thedetermination of the baseline, while the assumed positive effectof finding the true curved baseline is subordinated. By the samereasoning the worse performance using pre-processing method Gmight be explained.

The peak parameter based methods showed to be inferiorcompared to the multivariate methods most probably due to thefact that only a part of the wavenumber region was used in thesecases, using less of the available spectral information in thequantification procedure. Also when dealing with overlappingpeaks, the peak decomposition procedure may induce uncertain-ties in the determinations in peak parameters such as peakamplitudes and peak areas. However, the method of choice willdepend on the compound in question and unit operation underinvestigation, thus in certain cases can peak parameter analysiscontribute to sufficient information and reliable predictably(Rantanen et al., 2005).

In Appendix B expressions for spectral noise propagating intovariation of the determined amorphicity were developed whenapplying the method combination that was graded performingbest in this work, i.e. pre-processing method A combined withquantification by PCA. By using these expressions in a theoreticalanalysis, it was observed that due to spectral noise, the standarddeviation in determined amorphicity by the chosen pre-processingand quantification methods and the choice of wavenumber regionis expected to be around 0.74% for samples with low amorphouscontent. The standard deviation successively decreased forsamples with higher amorphous content down to 0.2% at pureamorphous sample. The observed difference in variances betweensamples containing high crystalline content and samples contain-ing high amorphous content is to a large extent a reflection of thePCA score (t-value) to fraction value transfer function (Eq. (A.6))and its derivative (Eq. (B.10)). The derivative is about three timeshigher at 0% amorphicity compared to 100% amorphicity, whichnearly agrees with the observed standard deviation ratios.

By putting spectral data from measurements of pure amor-phous and pure crystalline samples in Eqs. B.6-8 in Appendix B, it isshown that the spectral noise contribution to the total variation indetermined fraction amorphous is dominated by the impact of thebaseline. The impact of the baseline by the endpoints of thespectral interval was a factor 3 (amorphous) to a factor 6(crystalline) higher compared to the impact of spectral pointswithin the spectral interval.

The experimentally determined standard deviations of amor-phicity were around a factor 0.25 larger than the abovetheoretically obtained standard deviations both for the pureamorphous and pure crystalline samples. This discrepancy ofstandard deviation is due to unknown sources of error, but is mostprobable partly due to the fact that the baseline to some degree isapproximate, leading to additional variations in the normedspectra wherefrom the quantifications are made.


Knowing the sources of error and the magnitude of influence onthe variation in determined amorphicity gives a hint of where toput the effort if improved precision in the determinations arerequired. According to the relations derived in Appendix B, spectralnoise induced standard deviations of the determinations can bedecreased by increasing the intensity of the measured spectra bythe exposure time and/or exposure intensity in the measurementsand in the limit of infinite exposure in principle be reduced to zero.However considerations must be made that increasing theexposure might lead to changes of the sample and samplestructure e.g. by heating. The remaining error, due to the abovementioned unknown sources, is then close to the theoreticalminimum error, applying the chosen pre-processing and quantifi-cation methods and the choice of wavenumber interval. One way ofreducing them would, if the speculation above is correct, to furtherinvestigate and refine the preprocessing approaches. Investigatingfurther other quantification methods or wavenumber intervalmight also be a strategy of such issue. The relations derived inAppendix B could also be applied on any wavenumber region andany binary system applying the chosen preprocessing andquantification methods.

The magnitude of the variation of determined amorphicity forthe pure crystalline and amorphous samples is thus regarded asbeing due to the measuring and analysis procedure. Any largerobserved variation, such as clearly observed for the physicalmixtures, are here regarded being indications of additional sourcesof variation, such as heterogeneity in the samples. Indications oflarger variations, however smaller than for the physical mixtures,of determined amorphicity are also seen in the milled samplesindicating a small degree of heterogeneity also in these samples. Bythe small statistical basis, a more extensive investigation would berequired to determine the role of heterogeneity for the finalvariation of determined amorphous content

The effect of the sample heterogeneity is clearly illustrated by apre-trial series of Raman measurements made on physicalmixtures with the sample fixed, which resulted in remarkablyhigher variances of the determined fractions amorphous substance(data not shown) than on the series where the samples wererotated. In the rotating set up, the Raman signal sampling areabecame considerable larger, smearing out the heterogeneity by theaveraging effect. This observation also supports the use of the morefavorable and easily gained measuring set up by rotating thesample during the measurements.

4.2. Milled powders

Determination of the amount of apparent amorphous contentof each milled powder were made using all the pre-processing andquantification methods that are described in the method section(Table 7). Regarding the average of the determined percentage ofapparent amorphous content, a dependency of pre-processing andquantification methods used was notable and the effect of thechoice of pre-processing and quantification seemed more pro-nounced for the milled powders than for the physical mixtures.This discrepancy in obtained percentage values could be explainedby, that the relative intensity of different peaks obviously differsbetween milled samples and physical mixtures together with thequantification methods are considering different parts of thespectra differently for the percentage determination, see Fig. 8. Thelatter is most obvious in the peak parameter methods where one ortwo peaks are used while spectral values in the wavenumberregions covering the remainder of the peaks are not at allconsidered. Also in the PCA quantification this is the case, since theloadings are giving different weight at different wavenumbers.Nevertheless, a careful selection of the data handling procedure inthe spectrum analysis is critical, especially in the determination of

low levels of apparent amorphous content. As a generalisation, theRaman analysis gave indications of the apparent amorphicity of thesamples milled for 10, 300 and 1200 min of around 1%, 49% and 82%respectively.

The validation of the obtained apparent amorphous content forthe milled powders is problematic since the actual contents ofamorphous material of the milled particles are not known.However, the X-ray diffraction measurements gave supportingindications of the micro-structure of the bulk of the particles. Thediffraction pattern for powders milled for 10 min was similar to theoriginal powder but the intensity of the peaks was lower and it isconcluded that the shortest milling time gave a limited reductionin crystallinity of the particles. For powders milled for 300 min, afurther reduction in the intensity of the peaks together with slightbroadening was observed. This indicates that milling for 300 mingave a considerable increased degree of disorder in the particles.The diffraction pattern for powders milled for 1200 min finally wascharacterised by a halo with less distinct and further broadenedpeaks of low intensity. It is concluded that milling for 1200 mingave particles of high degree of disorder and the halo patternindicates presumable amorphous domains. Supported by thequalitative results of X-ray diffraction measurements, it thus couldbe concluded that the values of apparent amorphous contentobtained by Raman spectroscopy were reasonable.

Although amorphous particles are described in the literature aseither defective or partially amorphous there is in practice difficultto unambiguously differentiate between these two physicalnatures of amorphous material. It is argued (Chamarthy and Pinal,2008) that the processes of forming defects or fragmentingcrystallites precedes the formation of amorphous regions, i.e. asequential amorphisation process. Mechanistically this is de-scribed as an accumulation of defects or a reduction in size ofcrystallites and when the concentration of defects within a particleregion becomes high it will subsequently lead to the formation ofan amorphous phase. If a spatial distribution of amorphousdomains exists within a particle, a two-state model is applicable todescribe the physical nature of the particle. In addition, it has alsobeen argued (Pazesh et al., 2013) that an amorphous phase can beformed by a vitrification process during handing of powders due toparticle-particle collisions involving sliding and friction. Theamorphous phase thus formed will be peripherally located inthe particle. The exact physical nature of milled particles is thus anintricate issue and the representativeness of the calibrationprocedure used in Raman spectrum analysis involving referencesamples of physical mixtures can be questioned. In Fig. 8, Ramanspectra of milled samples were compared to synthesizedtheoretical spectra with the corresponding amorphous contentas the milled samples. The synthetized spectra generally capturedwell the overall profiles of the measured spectra for the milledsamples. The agreement was dependent on the wavenumberregion and the apparent amorphous content of the samples. For thelowest apparent amorphous content, the two spectra coincided asexpected, since the sample is almost entirely crystalline. For thehigher apparent amorphous content, the peaks numbered 2, 3, 4and 7, i.e. peaks with high relative intensity in the crystallinesample, more or less coincided with the corresponding physicalmixture, while the peaks numbered 5 and 6, i.e. peaks with highrelative intensity in the amorphous sample were merged anddeviated relative the corresponding physical mixture. It is howeverconcluded that by synthesising a spectrum using the crystallineand amorphous spectra, it is possible to obtain a reasonabledescription of the spectroscopic pattern of the milled material. Itseems natural to think that milling thus results in the formation ofamorphous regions and that the apparent amorphous content ofthe milled particles predominately is explained by the formation ofamorphous regions, located at the surface as a film of amorphous


material or dispersed within the particles as amorphous domains.However, there were observable differences between measuredand synthesized Raman spectra which may indicate that the natureof the milled particles cannot be described solely by a two-statesystem with amorphous and crystalline domains. The apparentamorphous content determined from Raman spectra of the milledparticles may thus represent an indication of amorphous domains,other detecting elements such as dislocation and crystalliteboundaries and transition boundaries between amorphous andcrystalline phases. The peak broadenings observed in the X-raydiffractograms may support that defects and fragmented crystal-lites existed in the milled particles. However, further amorphousresponse could occur if the crystallites or domains of order are sosmall that the diffraction features becomes too broad and weak tobe observed and/or not supporting phonon modes. Nevertheless,to better understand the disordering mechanisms of a crystallinesolid during milling, the use of low frequency Raman regions thatoffer a more sensitive probe of the lattice vibration of a crystallinestructure (Hédoux et al., 2011; Mah et al., 2015) could be ofpotential value.

To give a general advice regarding pre-processing andquantification methods would be questionable and most probablehighly dependent on the systems that are to be studied. Theintensity of fluorescence, the spectral outlook and the difference inspectra between the different constituents of the system may beparameters that are important for what methods actually will beoptimal for the quantification. The conclusions based on theexperiments done here are, to be strict, just applicable onamorphous/crystalline a-lactose system, however the methodsand design of investigation could be applied on other systems, anyfrequency region and with other spectroscopies, such as IR- andMicrowave spectroscopies. To be noted the here suggestedmethods relies on that the spectra of samples containing differentfractions of the two species could be described as linearcombinations of the spectra of the pure species. Therefore, caremust be taken how well this actually is applicable on the studiedsystem for the spectral method used and in the spectral range usedfor the quantification. If departures from linear combinations occurin the spectra of the mixed system, considerations if the heresuggested method approximate the mixed system well enough orif modifications of the method has to be done. The conclusionsfrom the theoretically derived equations on the other hand couldbe regarded as more general.

5. Conclusions

The result of the here done pre-processing and quantificationmethod comparison indicated that the two straight baseline pre-processing methods gave the best performance for lactose.Multivariate analysis methods generally performed superior tothe peak parameter methods.

By optimizing the experimental set-up, a precision in theestimate of amorphous content below 1% could be achieved. Theexperimental precision is slightly worse than could be expectedfrom theoretical calculations of propagated spectral noise. Thedevelopment of theoretical propagation of error expressions madeit possible to estimate the importance of difference noise sources,and this could be an aid if improvement of the precision of themeasurements are to be made in the here studied or in similarsystems. From the expressions it could also be read out theimportance of the baseline approaches for the variation ofdetermined amorphicity.

This study demonstrates that Raman spectroscopy proved to bean appropriate and effective technique in the quantification ofapparent amorphous content of milled lactose powder. This wasindicated by the achieved reasonable conformity between spectra

of milled sample and synthesized spectra with correspondingamorphous content.

Acknowledgments

Recipharm Pharmaceutical Development AB, Solna, Sweden isgratefully acknowledged for funding. The authors are grateful toProf. Peter Lazor, Department of Earth Sciences, MineralogyPetrology and Tectonics, Uppsala University, Sweden, for providingthe Raman instruments used in this research and for his helpfulcomments on an earlier draft. The authors wish to thank JoelHellrup, Department of Pharmacy, Uppsala University, Sweden forhelp with X-ray measurement and Göran Frenning, Department ofPharmacy, Uppsala University, Sweden for helpful discussion of thecalculation equations in appendixes.

Appendix A.

Derivation of equations used in the modified CLS analysis andderivation of a transformation equation from a scalar (e.g. PCA scorevalues, peak values or peak ratios) to fraction amorphous lactose

Due to the classical least squares method CLS spectrum, thespectrum vector yþ of a mixed sample is assumed to be the sum ofpure amorphous and pure crystalline spectrum vectors (y�þ

a andy�þc respectively) weighted by the fraction amorphous (fam) and

fraction crystalline lactose (1-fam) respectively (Haaland andThomas, 1988). The relation is expressed in Eq. (A.1):

yþ ¼ 1 � f amð Þ � yþ c þ f am � yþ a ðA:1ÞHowever, due to the normalization procedure in the pre-

processing, every spectrum was given the same integratedintensity, or corresponding norming quantity, in the wavenumberregion used for quantization. Dealing with normalized spectra, thecrystalline and amorphous spectra have to be multiplied by theirnorming quantities, Cc and Ca, respectively, if to be used in arelation like Eq. (A.1). To obtain a resulting spectrum that also isnormalized, the weighted sum has to be divided by the sum of theweights. The following expression is then obtained:

y ¼ ð1 � f amÞ � Cc � yc þ f am � Ca � yað1 � f amÞ � Cc þ f am � Ca

¼ ð1 � f amÞ � yc þ f am � C � yað1 � f amÞ þ f am � C

ðA:2Þ

where C¼Ca/Cc and ya, yc and y are the vector representation of thenormed amorphous and crystalline and sample spectra.

Having a sample set of known fractions of amorphous lactose,the ratio factor C could be determined by a non-linear fit, byminimizing the following expression:

Xnj

j¼1

Xni

i¼1

yj;i �ð1 � f am;jÞ � yc;i þ f am;j � C � ya;i

ð1 � f am;jÞ þ f am;j � C

!2

ðA:3Þ

where i runs over the wavenumbers in the spectral portion, j runsover the samples of different amorphous content, f am;j is thefractions of amorphous lactose in sample j, yj;i , yc;i and ya;i are thenormed spectral intensity at wavenumber i for sample j, for thecrystalline sample and the amorphous sample respectively.

Knowing C, a determination of f am could be obtained for anysample by a non-linear fit by minimizing the following expression:

Xni

i¼1

yi �1 � f amð Þ � yc;i þ f am � C � ya;i

1 � f amð Þ þ f am � C

� �2

ðA:4Þ


where yi is the normed spectral intensity at wavenumber i for thesample.

Applying an analogous reasoning as for the spectrum vectorsabove, any scalar quantity (r) varying with fam, such as a PCA score(t), peak parameters or from peak parameters derived quantities,could be expressed by fraction weighted sum of the scalar quantitydetermined in a pure crystalline and amorphous samplesrespectively. Those relations could thus be expressed as:

r ¼ 1 � f amð Þ � rc þ f am � Cr � ra1 � f amð Þ þ f am � Cr

ðA:5Þ

from which fam could be resolved as:

f am ¼ rc � rðCr � 1Þr � ðCr � ra � rcÞ ðA:6Þ

where rc and ra are the scalar quantities for the pure crystalline andpure amorphous samples respectively and Cr is the scalar ratiobetween the crystalline and amorphous spectra.

Having a sample set of known fractions of amorphous lactose,the ratio factor Cr could be determined by a non-linear fit, byminimizing the following expression:

Xni¼1

f am � rc � rðC � 1Þr � ðC � ra � rcÞ

� �2

ðA:7Þ

Appendix B.

Error analysis: Derivation of equation for propagating error ofmeasured spectral intensity into determined t- and percentage valuesin the case of linear baseline pre-processing (method A) followed byquantification by PCA

Denoting the measured spectral intensity by zi, where i iswavenumber index, the recalculated intensity after baselinecorrection and normalization of the spectra, yi is determinedaccording to:

yi ¼zi � zb;i

AðB:1Þ

where zb;i is the baseline z-coordinate and A is the area under thebaseline corrected spectrum.

With a linear background, zb;i could be expressed aszb;i¼ k � xiþzm, where xi is the wavenumber relative the center-wavenumber (405 cm�1) of the spectral portion under analysis, k isthe slope of the baseline and zm is the offset of the baseline at thecenter wavenumber. In turn, k and zm are given by:

zm ¼ ðznþz0Þ=2and

k ¼ zn � z0ð Þ=ðxn � x0Þ¼ðzn � z0Þ=Dxrange ðB:2Þwhere zn and z0 are the z-values at the upper (xn) and lower (x0)limits respectively of the spectral portion under analysis andDxrange ¼ xn � x0.

Further using the trapezoidal rule, A is given by:

A ¼ Sn�1

i¼1ðzi � zb;iÞ � Dx ¼ S

n�1

i¼1ðzi � k � xi � zmÞ

�Dx ¼ Sn�1

i¼1zi � Dx � k � Dx S

n�1

i¼1xi �

z0 þ zn2

� ðn � 1Þ

�Dx ¼ Sn�1

i¼1zi � Dx � z0 þ zn

2� ðn � 1Þ � Dx ðB:3Þ

where Dx = xi-xi-1. By the definition of the baselinez0 � zb;0 ¼ zn � zb;n ¼ 0, which has been used in the first equalityin Eq. (B.3). Further, by symmetry reasons

Pxi ¼ 0, which has been

used in the last equality in Eq. (B.3). Putting these expressions forarea and baseline into Eq. (B.1) then gives:

yi ¼zi � kxi � zm

Sn�1

i¼1zi � Dx � z0þzn

2 � ðn � 1Þ � Dx

ðB:4Þ

In PCA analysis, centered scaled values are used, i.e. bysubtracting the central spectrum value calculated asycent;i ¼ ðyc;iþya;iÞ=2, where yc;i and ya;i are the spectral value ofthe pure crystalline and pure amorphous sample respectively. Thescore (t) for the 1st principal component (PC1) has then beencalculated by weighting each centered values by correspondingloading, pi, for PC1, and then summing these values, i.e.:

t ¼ Sn�1

i¼1pi � yi � ycentr;i� � ¼ S

n�1

i¼1

pi � zi � zm � kxið ÞA

� Sn�1

i¼1pi � ycentr;i

¼Sn�1

i¼1pi � zi � zm � S

n�1

i¼1pi � k � S

n�1

i¼1pi � xi

A� S

n�1

i¼1pi � ycentr;i

¼Sn�1

i¼1pi � zi � k � S

n�1

i¼1pi � xi

A� S

n�1

i¼1pi � ycentr;i ðB:5Þ

The PCA is carried out on normalized and centered data, which

leads to thatXn�1

i¼1

pi ¼ 0, (see Appendix C) giving zm �Xn�1

i¼1

pi ¼ 0,

which in turn is used in the third equality in Eq. (B.5).

The variance of the score (t), s2t

Since terms with the products of loadings and spectral centervalues in the last sum in Eq. (B.5) are not changing between themeasurements, that sum will not contribute to the variance of t.

Thus only the term

Xn�1

i¼1

pi �zi�k�Xn�1

i¼1

pi �xi

A will give a contribution.

Denoting the numeratorXn�1

i¼1

pi � zi � k �Xn�1

i¼1

pi � xi ¼ N, the variance

of the score s2t , using Gauss approximation (Bevington and

Robinson, 1992), can be expressed as:

s2t

t2¼ s2

t

N2=A2 ¼ s2num

N2 þ s2area

A2 � 2CovðN; AÞ

N � A) s2

t ¼

N2

A2

s2num

N2 þ s2area

A2 � 2CovðN; AÞ

N � A

� �¼ s2

num

A2 þ N2 s2area

A4 � 2NCovðN; AÞ

A3

ðB:6Þwhere s2

num is the variance of numerator, s2area is the variance of the

area, and CovðN;AÞ is the covariance between the numerator andarea.

The variance of the numerator

Assuming that the variances, s2z;i, of the measured intensities

are equal for each spectral point we set s2z;i ¼ s2

z ¼ constant at allwavenumbers. The measured intensities could also be assumedbeing statistically independent. The relation k ¼ ðzn � z0Þ=Dxrangeis a weighted sum of zi-values which then gives:

s2k ¼ s2

n þ s20

Dx2Range¼ 2s2

z

Dx2Range


Using that also the whole numerator can be seen as a weightedsum of zi-values the following is obtained:

s2num ¼ S

n�1

i¼1p2i � s2

z;i þ s2k S

n�1

i¼1pi � xi

!2

¼ s2z � S

n�1

i¼1p2i þ

2s2z

Dx2rangeSn�1

i¼1pi � xi

!2

ðB:7Þ

The variance of the area (denominator)

In a similar way, using that the expression for the area also canbe seen as a weighted sum of zi-values, (see Eq. (B.4)), the varianceof the denominator s2

area is given by

s2area ¼ S

n�1

i¼1s2z;i � Dxð Þ2 þ s2

z;0

22 þ s2z;n

22

!� ðn � 1Þ2 � ðDxÞ2 ¼

ðn � 1Þ � s2z � Dxð Þ2 þ s2

z

2� ðn � 1Þ2 � ðDxÞ2

ðB:8Þ

The covariance between numerator and area

Assuming that the intensity observations at different wave-numbers are statistically independent, the following must be validregarding the covariances: Cov zi;zj

� � ¼ di;js2z and Covðz0 þ zn;zn �

z0Þ ¼ 0This gives:

CovðN; AÞ ¼ Cov Sn�1

i¼1pi � zi

!� zn � z0

Dxrange� Sn�1

i¼1pi � xi

!" #;

Sn�1

i¼1zj � Dx

!� z0 þ zn

2� n � 1ð ÞDx

� �" #!

¼ Cov Sn�1

i¼1pi � zi

!; S

n�1

i¼1zj � Dx

! !

þ Cov Sn�1

i¼1pi � zi

!;

z0 þ zn2

� n � 1ð ÞDx� � !

þ Covzn � z0Dxrange

� Sn�1

i¼1pi � xi

!; S

n�1

i¼1zj � Dx

! !

þ Covzn � z0Dxrange

�Xn�1

i¼1

pi � xi

!;

z0 þ zn2

� n � 1ð ÞDx� � !

¼ Cov Sn�1

i¼1pi � zi

!; S

n�1

i¼1zj � Dx

! !þ 0 þ 0 þ 0½ �

¼ Sn�1

j¼1Sn�1

i¼1Cov pi � zi; zjDx

� � ¼ Dx Sn�1

i¼1pi � s2

z ¼ 0

ðB:9Þ

Recalling thatXn�1

i¼1

pi ¼ 0, the last equality is obtained.

The variance of fraction amorphous lactose determined

According to Appendix A, the fraction amorphous fam isobtained from the score t as:

f am ¼ tc � tðC � 1Þt � ðC � ta � tcÞ;

where r has been replaced by t in Eq. (A.6). This gives:

df amdt

¼ C � ðta � tcÞCðt � taÞ � ðt � tcÞ½ �2

ðB:10Þ

and Gauss approximation gives in turn:

sf ¼df amdt

st ¼ C � ðta � tcÞCðt � taÞ � ðt � tcÞ½ �2

st ðB:11Þ

where s2f is the variance of the determined fraction amorphous

lactose.

Appendix C.

Showing that the sum of scores equals zero when PCA is performed oncentered values of normalized spectra

yc;centr ¼ yc �yc þ ya

2¼ yc � ya

2ðC:1Þ

ya;centr ¼ ya �yc þ ya

2¼ �yc � ya

2ðC:2Þ

Summing the individual centered spectral intensity valuesgives:

Xn�1

i¼1

yc;centr;i ¼12

Xn�1

i¼1

yc;i � ya;i ¼12

Xn�1

i¼1

yc;i �Xn�1

i¼1

ya;i

!

¼ 12� ð1 � 1Þ ¼ 0 ðC:3Þ

since by norming bothXn�1

i¼1

yc;i ¼ 1 andXn�1

i¼1

ya;i ¼ 1

The scores, pi, are the direction cosines of PC1 relative thewavenumber axes, yi. Since the PCA is only carried out on spectra ofthe pure amorphous and pure crystalline samples, PC1 is the axiscrossing the averages of pure amorphous spectra and purecrystalline spectra respectively. Thus:

pi ¼yc;centr;iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn�1

i¼1

y2c;centr;i

vuut)Xn�1

i¼1

pi ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn�1

i¼1

y2c;centr;i

vuutXn�1

i¼1

yc;centr;i

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn�1

i¼1

y2c;centr;i

vuut� 0 ¼ 0 ðC:4Þ

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